224 XIII INTEGRATION inequality in the reverse direction (with equality when h e &*(X x Y)) for lower integrals: (13.21.4.3) IT /z(x, y) dl(x) dn(y) g f dl(x) !h(x, y) dfi(y). (13.21.5) 7/*N is a v-negligible subset ofXx Y, then the set of points x e X such that the section N(^) c: Y of N is not ^-negligible is ^-negligible (in other words, we have ju(N(.x)) = 0 almost everywhere with respect to A). This follows immediately from (13.21.4.2) applied to h = <pN. (13.21 .6) If his a ^-measurable mapping ofX x Y into a topological space E, then the set of points xeX for which the partial mapping yt-*h(x, y) is not ^-measurable is ^-negligible (in other words, the mapping y\-*h(x9y) is ju- measurable almost everywhere with respect to A). By hypothesis, there exists a partition of X x Y consisting of a v-negligible set N and a sequence (Kn)B^i of compact sets, such that each of the restric- tions A|KB is continuous (13.9). Let M be the A-negligible set of points ;ceX at which the section N(x) of N is not ju-negligible (13.21.5). For each x $ M, Y admits a partition consisting of compact sets Kn(x) (n ^ 1) and the /^-negligible set N(x), such that the restriction of y\->h(x, y) to each of the sets KB(#) is continuous. Hence the result. It should be noted that it can happen that for each x e X the function y*-*f(x,y) is ^-measurable, and for each y e Y the function x\-+f(x,y) is A-measurable, but that f is not v-measurable. (1 3.21 .7) (Lebesgue-Fubini Theorem) Let 1, \i be positive measures on X and Y, respectively, and v = A ® \i their product. For each \-integrable map- ping h o/X x Y into R, the set of points x e X such that the partial mapping yt-*h(x,y) is not ^-integrabte is ^-negligible; the function XH-> \h(x, o ' J which is defined almost everywhere with respect to A, is X-integrable ; and (12.21.7.1) h(x, y) dl(x It follows from (13.21.4.1) that the function xi~» f*|A(#, y)\ dfj,(y) is finite on the complement of a A-negligible set Nx (13.6.4). On the other hand, the set N2 of points x e X such that y H> h(x, y) is not ju-measurable is A-negligibleion of the (finitely many) interiors of sets VjO^jCj)) which contain y; the